| In the first part of this dissertation, we present two models for image reconstruction and decomposition, based on ideas of Rudin-Osher-Fatemi bounded total variation regularization [57], and on Y. Meyer's ideas of using distributional normed spaces for modeling oscillatory functions [46]. Given a degraded image ƒ = Ku + n, where K is a blurring operator and n represents additive noise, to extract a clean image u, we consider variational problems of the form inf {lcub}E(u) := lambdaƒ1( u) + F2 (ƒ-Ku){rcub}, where F1 and F2 are functionals acting on some normed spaces. Inspired by Meyer and motivated by Mumford-Gidas [49] and Osher-Sole-Vese [52], we propose for the first model F1 (u) =∫ | Du|, the total variation of u, and F 2 (ƒ - Ku) = ||ƒ - Ku|| H-s, the norm on the Hilbert-Sobolev space of negative exponent. For the second model, we consider for F1 a general regularization penalty phi(u) = ∫&phis;(|Du |)dx < infinity, where &phis; is positive, increasing, and grows at most linearly at infinity, while for F 2, we propose the dual penalty F2(ƒ - Ku) = phi* (ƒ - Ku). We present algorithms for computing the dual phi* and for solving the minimization problem inf{lcub}E(u) := lambdaphi)( u) + phi*(ƒ - Ku){rcub}. We also present theoretical and numerical results. In the particular case when 4) is the total variation of u, we show that the proposed model recovers the (BV, BV*) decomposition, as in Y. Meyer's model.; In the second part of the dissertation, we present a solution to the problem of edge detection in streak-camera images, which arises from scientific experiments. We present an adaptation of the Chan-Vese segmentation model [18]: a novel approach in which we segment two dimensional images by a 1D model.; Numerical results from each model will be presented in each part. |
